How Can ZFC/PA do much of Math - it Can't Even Prove PA is Consistent (EASY PROOF)
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From: herbzet on 3 Jul 2010 01:57 MoeBlee wrote: > So whether we adopt: > > consistent <-> [does not] derive a contradiction > or > consistent <-> there's a formula not derivable > > would determine how the paraconsistent advocate would couch his views > as to consistency, as he would not mind a theory with contradictions > as long as the "explosive property" were absent so that contradictions > don't derive everything. > > I think we are in accord, as far as I can tell. Yeah, I think. A paraconsistent advocate would call an inconsistent theory (derives a contradiction) "inconsistent" but wouldn't care, so long as the theory isn't also trivial. That is, the convention in usage seems to be that inconsistency does not imply triviality. -- hz
From: Charlie-Boo on 3 Jul 2010 07:20 On Jun 29, 10:22 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > Charlie-Boo <shymath...(a)gmail.com> writes: > > What is that formal expression? > > To find out you need to read a logic book. Then why don't you (anyone) name one that has it? You can't name a book that has the proof or even the theorem spelled out, and you can't give it yourself, or even an outline. Yet you claim it can be done. Is that Mathematics? I thought statements had to be proven in Mathematics. C-B > It appears the generous > explanations various people have provided for your benefit in news are > not sufficient. > > -- > Aatu Koskensilta (aatu.koskensi...(a)uta.fi) > > "Wovon man nicht sprechan kann, dar ber muss man schweigen" > - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Charlie-Boo on 3 Jul 2010 07:22 On Jun 29, 10:16 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > Charlie-Boo <shymath...(a)gmail.com> writes: > > Ok. What is the reference to the proof in ZFC of PA consistency doing > > it that way? > > It's in Shelah's _Cardinal Arithmetic_ p.3245 - 4325238532. Basically, > you just do a triple-fold transfinite recursion over a coherent extender > sequence to obtain a suitable premouse, and iterate the upward Mostowski > collapsing lemma a few times. To remove the extendible cardinal > introduce some Aronszjan trees using Sacks forcing. Where's the part about how ZFC is used to do it? C-B > -- > Aatu Koskensilta (aatu.koskensi...(a)uta.fi) > > "Wovon man nicht sprechan kann, darüber muss man schweigen" > - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Charlie-Boo on 3 Jul 2010 07:26 On Jun 29, 9:16 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > billh04 <h...(a)tulane.edu> writes: > > Are you saying that it is a theorem of ZFC that PA is consistent? > > Sure. That is, the statement "PA is consistent" formalized in the > language of set theory as usual is formally derivable in ZFC (and No it isn't. PA can't do it and ZFC has nothing outside of PA relevant to the proof. Now, I refuted it - either prove it can be done, or refute my refutation, or retract the statment, ok? Isn't that how Mathematics works? C-B > already in much weaker theories). > > -- > Aatu Koskensilta (aatu.koskensi...(a)uta.fi) > > "Wovon man nicht sprechan kann, darüber muss man schweigen" > - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Charlie-Boo on 3 Jul 2010 07:28
On Jun 29, 9:13 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > Charlie-Boo <shymath...(a)gmail.com> writes: > > ZFC was designed to avoid paradoxes by making explicit what can be a > > set. It doesn't do anything else except what the Peano Axioms give > > it. > > Does PA give us Borel determinacy? Is Borel determinacy trivial? Prove ZFC can prove it and PA can't. C-B > -- > Aatu Koskensilta (aatu.koskensi...(a)uta.fi) > > "Wovon man nicht sprechan kann, darüber muss man schweigen" > - Ludwig Wittgenstein, Tractatus Logico-Philosophicus |